I have the polynomail f(x)=x^5 +2x+2 over rationals. Let M be teh splitting field of this over rationals.
Then I want to show that the galois group is S_5.
My argument is as follows:
f(x) is irreducible by Eisenstein with p=2. and the galois group is isomorphic to a subgroup of S_5, let us call it H.
we have that the order of H is divisible by 5, then by using Cauchy's theorem we have that there is an element of order 5 in H, so it must be a 5-cycle. So H contains a five cycle.
Now I can show that f has only one real root?
Then what will be the argument be? I want to conclude something like that H contains a transposition, so H must be S_5.
Any help, any hints? Thanks.